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Theorem fsuppco2 8463
Description: The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 8464 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
fsuppco2.z (𝜑𝑍𝑊)
fsuppco2.f (𝜑𝐹:𝐴𝐵)
fsuppco2.g (𝜑𝐺:𝐵𝐵)
fsuppco2.a (𝜑𝐴𝑈)
fsuppco2.b (𝜑𝐵𝑉)
fsuppco2.n (𝜑𝐹 finSupp 𝑍)
fsuppco2.i (𝜑 → (𝐺𝑍) = 𝑍)
Assertion
Ref Expression
fsuppco2 (𝜑 → (𝐺𝐹) finSupp 𝑍)

Proof of Theorem fsuppco2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fsuppco2.g . . . 4 (𝜑𝐺:𝐵𝐵)
2 ffun 6188 . . . 4 (𝐺:𝐵𝐵 → Fun 𝐺)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐺)
4 fsuppco2.f . . . 4 (𝜑𝐹:𝐴𝐵)
5 ffun 6188 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
64, 5syl 17 . . 3 (𝜑 → Fun 𝐹)
7 funco 6071 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
83, 6, 7syl2anc 565 . 2 (𝜑 → Fun (𝐺𝐹))
9 fsuppco2.n . . . 4 (𝜑𝐹 finSupp 𝑍)
109fsuppimpd 8437 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11 fco 6198 . . . . 5 ((𝐺:𝐵𝐵𝐹:𝐴𝐵) → (𝐺𝐹):𝐴𝐵)
121, 4, 11syl2anc 565 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐵)
13 eldifi 3881 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 6417 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
154, 13, 14syl2an 575 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssid 3771 . . . . . . . 8 (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)
1716a1i 11 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
18 fsuppco2.a . . . . . . 7 (𝜑𝐴𝑈)
19 fsuppco2.z . . . . . . 7 (𝜑𝑍𝑊)
204, 17, 18, 19suppssr 7477 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2120fveq2d 6336 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
22 fsuppco2.i . . . . . 6 (𝜑 → (𝐺𝑍) = 𝑍)
2322adantr 466 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 𝑍)
2415, 21, 233eqtrd 2808 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 𝑍)
2512, 24suppss 7476 . . 3 (𝜑 → ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
26 ssfi 8335 . . 3 (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐺𝐹) supp 𝑍) ∈ Fin)
2710, 25, 26syl2anc 565 . 2 (𝜑 → ((𝐺𝐹) supp 𝑍) ∈ Fin)
28 fsuppco2.b . . . . 5 (𝜑𝐵𝑉)
29 fex 6632 . . . . 5 ((𝐺:𝐵𝐵𝐵𝑉) → 𝐺 ∈ V)
301, 28, 29syl2anc 565 . . . 4 (𝜑𝐺 ∈ V)
31 fex 6632 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑈) → 𝐹 ∈ V)
324, 18, 31syl2anc 565 . . . 4 (𝜑𝐹 ∈ V)
33 coexg 7263 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3430, 32, 33syl2anc 565 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
35 isfsupp 8434 . . 3 (((𝐺𝐹) ∈ V ∧ 𝑍𝑊) → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
3634, 19, 35syl2anc 565 . 2 (𝜑 → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
378, 27, 36mpbir2and 684 1 (𝜑 → (𝐺𝐹) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  cdif 3718  wss 3721   class class class wbr 4784  ccom 5253  Fun wfun 6025  wf 6027  cfv 6031  (class class class)co 6792   supp csupp 7445  Fincfn 8108   finSupp cfsupp 8430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-supp 7446  df-er 7895  df-en 8109  df-fin 8112  df-fsupp 8431
This theorem is referenced by:  gsumzinv  18551  gsumsub  18554
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